Is there a model number on the A-K condenser? I suspect it's for the broadcast band. I would open it up and determine just what is inside and where the connections go to the outside. Do we know it's a condenser and not a variometer? There's that tube socket on the back which doesn't look standard and looks removeable.
Here's an experiment: use test jumpers and haywire up a crystal set with the condenser and the long winding of each coil. All you need is the jumpers, a germanium diode, and some high impedance headphones. If you don't have the latter, you can use a powered computer speaker with its built-in amp. Best with an outdoor long wire but if you don't have that string out 20 feet or more of wire wherever you can. This might just tune the broadcast band. If you pick up a station, listen for its frequency and note how far open the condenser plates are, and compare with an operating radio tuned to that station. If they are about the same angle then you're in business.
Edit: adding the schematics below. First one is with headphones. Second is if you are using an external amp like a computer speaker/amp. R can be anything from 10K to 50K ohms, higher the better, C can be from .01 to .047 mfd. H goes to the high side of the amp input (center conductor on a shielded cable) and S goes to the shield side. Turn up the volume on the amp. Notice that the set uses a direct ground connection so don't do this with an AC/DC amp unless you use an isolation transformer. A computer speaker/amp will be OK.
In both cases ignore that the antenna is tapped on the coil and connect the antenna to the top of the coil along with the stator of the condenser and the anode of the crystal. The cathode of the crystal goes towards the right and is the band painted on the crystal, although it will work whichever way around you install it.
Do you have a signal generator? Could also use that loosely coupled to the coil and sweep all over the place searching for the resonant frequency range.
